Chi-Square Genetics Calculator

The question the test answers

Real crosses never give perfect ratios. If a cross expected to give 3 : 1 produces 82 and 18 offspring, is that close enough? The chi-square goodness-of-fit test measures how far observed counts are from the expected counts and asks how often chance alone would produce a difference that large. The statistic is χ² = Σ (observed − expected)² ÷ expected, summed over the classes. The degrees of freedom are the number of classes minus one. The p-value is the probability of a χ² at least that large if the expected ratio is correct.

Mendel’s peas

Mendel’s dihybrid cross of round yellow and wrinkled green peas produced 315 round yellow, 108 round green, 101 wrinkled yellow and 32 wrinkled green seeds, 556 in all. For a 9 : 3 : 3 : 1 ratio the expected counts are 312.75, 104.25, 104.25 and 34.75. The contributions to χ² are 0.016, 0.135, 0.101 and 0.218, a total of 0.470 with 3 degrees of freedom. The p-value is 0.925: a fit this good or better happens by chance only about 7.5% of the time. For his monohybrid seed-shape cross, 5,474 round and 1,850 wrinkled, χ² is 0.263 against 3 : 1, p = 0.608. The statistician R. A. Fisher noted in 1936 that Mendel’s results were collectively closer to expectation than chance would usually allow, a debate that continues; the arithmetic is a good reminder that too good a fit is also informative.

Reading the p-value

By convention, p below 0.05 is called significant: the deviation is unlikely to be chance alone, and the hypothesised ratio is rejected. For 1 degree of freedom the critical value is 3.84, for 2 it is 5.99 and for 3 it is 7.81. A common misconception is that p above 0.05 proves the ratio is correct. It does not; it only means the data do not contradict it, and a small sample may fail to detect a real difference. Another mistake is to run the test on percentages instead of counts: χ² depends on sample size, and 82% versus 75% means something very different for 100 offspring than for 10,000.

When the test is unreliable

The chi-square distribution is an approximation that works when every expected count is at least about 5. With smaller expected counts, combine classes or use an exact test. The classes must be mutually exclusive and counted, not measured, and each offspring must be independent. If allele frequencies or other parameters are estimated from the same data, as in a Hardy-Weinberg test, subtract an extra degree of freedom for each.

How to use it

1. Enter the observed counts for each phenotype class, separated by commas.
2. Enter the expected ratio with one term per class, such as 9:3:3:1.
3. Read the expected counts, each class’s contribution, χ², the degrees of freedom and the p-value.
4. Check that every expected count is at least 5 before trusting the result.